CAAC 2022 Schedule
Dates: Friday, Saturday, Sunday January 21,22 and 23. All posted times are Eastern time zone.
Fri 2pm - 2:50pm - Allen Knutson
(Note: this talk will be a joint presentation with the CSMQ Colloque)
Fri 3pm - 3:20pm - Laura Colmenarejo
Virtual Coffee Break (3:20-3:50)
Fri 3:50pm - 4:10pm - Colleen Robichaux
Fri 4:20pm - 4:40pm - Emmanuel O Neye
Sat 9:30am - 10:20am - Laura Escobar
Sat 10:30am - 10:50am - Selvi Kara
Virtual Coffee Break (10:50-11:20)
Sat 11:20am - 11:40am - Ayah Almousa
Sat 11:50am - 12:10pm - Ben Dequêne
Lunch break (12:10pm-1:30pm)
Sat 1:30pm - 1:50pm - Vasu Tewari
Sat 2:00pm - 2:20pm - Gidon Orelowitz
Sat 2:30pm - 3:20pm - Eloísa Grifo
Sat 3:20pm - discussion
Sun 9:30am - 10:20am - Oliver Pechenik
Sun 10:30am - 10:50am - Hugh Thomas
Virtual Coffee Break (10:50-11:10)
Sun 11:10am - 12:00pm - Colin Ingalls
Invited Speakers
Laura Escobar (Washington University in St. Louis)
Title: Which Schubert varieties are Hessenberg varieties?
Eloísa Grifo (University of Nebraska-Lincoln)
Title: Symbolic powers
Colin Ingalls (Carleton University)
Title: Some Matrix Factorizations of Discriminants
Allen Knutson (Cornell University)
Title: The commuting variety and generic pipe dreams
Oliver Pechenik (University of Waterloo)
Title: Castelnuovo-Mumford regularity of matrix Schubert varieties
Contributed Talks
Ayah Almousa (University of Minnesota)
Title: Triangulations of root polytopes and polarizations of monomial ideals
Laura Colmenarejo (NCSU)
Title: Chromatic symmetric functions of Dyck paths and q-rook theory
Benjamin Dequêne (UQAM)
Title: Jordan recoverability of some subcategories of modules over gentle algebras
Selvi Kara (University of Utah)
Title: Multi-Rees Algebras of Strongly Stable Ideals
Emmanuel O Neye (University of Saskatchewan)
Title: A Grobner Basis for Schubert Patch Ideals
Gidon Orelowitz (University of Illinois, Urbana-Champaign)
Title: The Kostka semigroup and its Hilbert basis
Colleen Robichaux (UIUC)
Title: Castelnuovo-Mumford regularity of ladder determinantal ideals via Grothendieck polynomials
Vasu Tewari (University of Hawaii at Manoa)
Title: Polynomials modulo quasisymmetric polynomials and the class of the permutahedral variety
Hugh Thomas (UQAM)
Title: An analogue of the associahedron for finite-dimensional algebras
Titles and Abstracts
Ayah Almousa (University of Minnesota)
Title: Triangulations of root polytopes and polarizations of monomial ideals
Abstract: We discuss connections between triangulations of root polytopes and polarizations of certain monomial ideals. We show that every triangulation of a root polytope gives rise to a polarization and discuss implications of this fact in algebra and combinatorics.
Laura Colmenarejo (NCSU)
Title: Chromatic symmetric functions of Dyck paths and q-rook theory
Abstract: Given a graph and a set of colors, a coloring is a function that associates each vertex in the graph with a color. In 1995, Stanley generalized this definition to symmetric functions by looking at the number of times each color is used and extending the set of colors to ℤ+.In 2012, Shareshian and Wachs introduced a refinement of the chromatic functions for ordered graphs as q-analogues.
In the particular case of Dyck paths, Stanley and Stembridge described the connection between chromatic symmetric functions of abelian Dyck paths and square hit numbers, and Guay-Paquet described their relation to rectangular hit numbers. Recently, Abreu-Nigro generalized the former connection for the Shareshian-Wachs q-analogue, and in unpublished work, Guay-Paquet generalized the latter.
In this talk, I want to give an overview of the framework and present another proof of Guay-Paquet's identity using q-rook theory. This is recent work with Alejandro H. Morales and Greta Panova.
Benjamin Dequêne (UQAM)
Title: Jordan recoverability of some subcategories of modules over gentle algebras
Abstract: Gentle algebras form a class of finite dimensional algebras introduced by Assem and Skowroński in the 80’s. Indecomposable modules over such an algebra admit a combinatorial description in terms of strings and bands, which are walks in the associated gentle quiver (satisfying some further conditions), thanks to work of Butler and Ringel. A subcategory C of modules is said to be Jordan recoverable if a module X in C can be recovered from the Jordan forms, at each vertex, of a generic nilpotent endomorphism. This data is encoded by a tuple of integer partitions.
After we have introduced some definitions and set the context, the main aim of the talk is to explain the notion of Jordan recoverability through various examples, and to highlight a combinatorial characterization of when that property holds for some special subcategories of modules. This result is extends the work of Garver, Patrias and Thomas in Dynkin types. If time allows, we may discuss some open questions related to this result and, in particular, exhibit new ideas to characterize all the subcategories of modules that are Jordan recoverable in the A_n case.
Laura Escobar (Washington University in St. Louis)
Title: Which Schubert varieties are Hessenberg varieties?
Abstract: Schubert varieties and Hessenberg varieties are subvarieties of the flag variety with connections to both algebraic combinatorics and representation theory. I will discuss joint work with Martha Precup and John Shareshian in which we investigate which Schubert varieties are Hessenberg varieties. In the process, we give a restriction on the Euler characteristic of codimension-one Hessenberg varieties, as well as a description of the singular loci of regular codimension-one Hessenberg varieties.
Eloísa Grifo (University of Nebraska-Lincoln)
Title: Symbolic powers
Abstract: We will give an overview of symbolic powers, a classical topic in commutative algebra full of easy to state open questions, highlighting problems of a combinatorial flavor.
Selvi Kara (University of Utah)
Title: Multi-Rees Algebras of Strongly Stable Ideals
Abstract: In this talk, we will focus on Rees and multi-Rees algebras of strongly stable ideals. In particular, we will discuss the Koszulness of these algebras through a systematic study of these objects via three parameters: the number of ideals, the number of Borel generators of each ideal, and the degrees of Borel generators. In addition, we utilize combinatorial objects such as fiber graphs to detect Gröbner bases and Koszulness of these algebras.
Allen Knutson (Cornell University)
Title: The commuting variety and generic pipe dreams
Abstract: Nobody knows whether the scheme "pairs of commuting nxn matrices" is reduced. I'll show how this scheme relates to matrix Schubert varieties, and give a formula for its equivariant cohomology class (and that of many other varieties) using "generic pipe dreams" that I'll introduce. These interpolate between ordinary and bumpless pipe dreams. With those, I'll rederive both formulae (ordinary and bumpless) for double Schubert polynomials. This work is joint with Paul Zinn-Justin.
Colin Ingalls
(Carleton University)
Title:
Some Matrix Factorizations of Discriminants
Abstract: Higher Specht polynomials give a basis of the polynomial ring that respects the decomposition into irreducible representations of the symmetric group. We use these polynomials to construct matrix factorizations of the discriminant indexed by partitions. The Cohen-Macaualy modules associated to these matrix factorizations give a noncommutative resolution of the discriminant. We also discuss extensions of these results to other complex reflection groups. This is joint work with Eleonore Faber, Simon May and Marco Talarico.
Emmanuel O Neye (University of Saskatchewan)
Title: A Grobner Basis for Schubert Patch Ideals
Abstract: Schubert patch ideals are a class of generalized determinantal ideals. They are prime defining ideals of open patches of Schubert varieties in the type A flag variety. In this talk, E. Gorla, J. Migliore, and U. Nagel’s “Grobner basis via linkage” technique will be adapted to prove a conjecture of A. Yong, namely, the essential minors of every Schubert patch ideal form a Grobner basis. Using the same approach, the result of A. Woo and A. Yong that the essential minors of a Kazhdan-Lusztig ideal form a Grobner basis will be recovered. With respect to standard grading, it will be shown that the homogeneous Schubert patch ideals and homogeneous Kazhdan-Lusztig ideals (and hence, the Schubert determinantal ideals) are glicci.
Gidon Orelowitz (University of Illinois, Urbana-Champaign)
Title: The Kostka semigroup and its Hilbert basis
Abstract: The Kostka semigroup consists of pairs of partitions with at most r parts that have positive Kostka coefficient. For this semigroup, Hilbert basis membership is an NP-complete problem. We introduce KGR graphs and conservative subtrees, through the Gale-Ryser theorem on contingency tables, as a criterion for membership. In our main application, we show that if a partition pair is in the Hilbert basis then the partitions are at most r wide.
Oliver Pechenik (University of Waterloo)
Title: Castelnuovo-Mumford regularity of matrix Schubert varieties
Abstract: Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo-Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo-Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo-Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo-Mumford polynomials agree up to scalar multiple. The degree of the Grothendieck polynomial is a new permutation statistic which we call the Rajchgot index; we develop the properties of Rajchgot index and relate it to major index and to weak order.
Colleen Robichaux (University of Illinois, Urbana-Champaign)
Title: Castelnuovo-Mumford regularity of ladder determinantal ideals via Grothendieck polynomials
Abstract: We give a degree formula for Grothendieck polynomials indexed by vexillary permutations. We apply this formula to compute the Castelnuovo-Mumford regularity for certain classes of generalized determinantal ideals. In particular, we give a combinatorial formula for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of certain Kazhdan-Lusztig ideals, including those coming from open patches of Grassmannians. This provides a correction to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri (2015). This is joint work with Jenna Rajchgot and Anna Weigandt.
Vasu Tewari (University of Hawaii at Manoa)
Title: Polynomials modulo quasisymmetric polynomials and the class of the permutahedral variety
Abstract: I will talk about a new basis of the polynomial ring that refines Schubert polynomials. This basis is inspired by work of Aval-Bergeron-Bergeron in the context of reducing polynomials modulo the ideal of positive degree quasisymmetric polyomials. We use this to find a manifestly nonnegative integral expansion for the class of the permutahedral variety in terms of Schubert classes, with a novel parking procedure playing a key role.
Hugh Thomas (UQAM)
Title: An analogue of the associahedron for finite-dimensional algebras
Abstract: I will describe an algebraic variety associated to an algebra of finite representation type. Starting from an orientation of a type A_n quiver, one gets a curvy version of the associahedron, while in general one gets a variety whose totally positive part encodes the tau-tilting theory of the algebra. Many mysteries about this construction remain, including how to define a similar variety when the algebra is not of finite representation type. This is part of joint work with Nima Arkani-Hamed, Hadleigh Frost, Pierre-Guy Plamondon, and Giulio Salvatori. Part of the motivation comes from string theory, which I will attempt to explain if I have time.